In this paper a mathematical model is presented that describes growth, immune escape, and siRNA treatment of tumors. The model consists of a system of nonlinear, ordinary differential equations describing tumor cells and immune effectors, as well as the immuno-stimulatory and suppressive cytokines IL-2 and TGF-$\beta$. TGF-$\beta$ suppresses the immune system by inhibiting the activation of effector cells and reducing tumor antigen expression. It also stimulates tumor growth by promoting angiogenesis, explaining the inclusion of an angiogenic switch mechanism for TGF-$\beta$ activity. The model predicts that increasing the rate of TGF-$\beta$ production for reasonable values of tumor antigenicity enhances tumor growth and its ability to escape host detection. The model is then extended to include siRNA treatment which suppresses TGF-$\beta$ production by targeting the mRNA that codes for TGF-$\beta$, thereby reducing the presence and effect of TGF-$\beta$ in tumor cells. Comparison of tumor response to multiple injections of siRNA with behavior of untreated tumors demonstrates the effectiveness of this proposed treatment strategy. A second administration method, continuous infusion, is included to contrast the ideal outcome of siRNA treatment. The model's results predict conditions under which siRNA treatment can be successful in returning an aggressive, TGF-$\beta$ producing tumor to its passive, non-immune evading state.

A mathematical model is developed that investigates polyclonality and decreased apoptosis as mechanisms for the androgen-independent relapse of human prostate cancer. The tumor is treated as a continuum of two types of cells (androgen-dependent and androgen- independent) whose proliferation and apoptotic death rates differ in response to androgen rich and androgen poor conditions. Insight into the tumor's response to therapies which both partially and completely block androgen production is gained by applying a combination of analytical and numerical techniques to the model equations. The analysis predicts that androgen deprivation therapy can only be successful for a small range of the biological parameters no matter how completely androgen production is blocked. Numerical simulations show that the model captures all three experimentally observed phases of human prostate cancer progression including exponential growth prior to treatment, androgen sensitivity immediately following therapy, and the eventual androgen-independent relapse of the tumor. Simulations also agree with experimental evidence that androgen-independent relapse is associated with a decrease in apoptosis without an increase in proliferation.

The tumor suppressor p53 protein shows various dynamic responses depending on the types and extent of cellular stresses. In particular, in response to DNA damage induced by γ-irradiation, cells generate a series of p53 pulses. Recent research has shown the importance of sustaining repeated p53 pulses for recovery from DNA damage. However, far too little attention has been paid to understanding how cells can sustain p53 pulses given the complexities of genetic heterogeneity and intrinsic noise. Here, we explore potential molecular mechanisms that enhance the sustainability of p53 pulses by developing a new mathematical model of the p53 regulatory system. This model can reproduce many experimental results that describe the dynamics of p53 pulses. By simulating the model both deterministically and stochastically, we found three potential mechanisms that improve the sustainability of p53 pulses: 1) the recently identified positive feedback loop between p53 and Rorα allows cells to sustain p53 pulses with high amplitude over a wide range of conditions, 2) intrinsic noise can often prevent the dampening of p53 pulses even after mutations, and 3) coupling of p53 pulses in neighboring cells via cytochrome-c significantly reduces the chance of failure in sustaining p53 pulses in the presence of heterogeneity among cells. Finally, in light of these results, we propose testable experiments that can reveal important mechanisms underlying p53 dynamics.

Considerable research is aimed at determining the mechanisms by which hormone-refractory prostate cancer develops. In an effort to assist in the understanding of recurrent prostate cancer and the cellular processes that mediate this disease, a mathematical model is presented that describes both the pretreatment growth and the posttherapy relapse of human prostate cancer xenografts. Our goal is to evaluate the interplay between the multiple mechanisms that have been postulated as causes of androgen-independent relapse. Simulations of the model show that molecular events that render the androgen receptor irrelevant to disease progression, such as upregulation of BCL2, can result in relapse after androgen deprivation therapy. However, decreased apoptosis of androgen-independent cells alone overestimates the effects of hormone therapy when compared to experimental data. When decreased apoptosis is combined with continual androgen receptor activation, the posttherapy growth dynamics are in excellent correlation with experimental observations of the growth of LuCaP xenografts. Furthermore, the mathematical model predicts that upregulation of the androgen receptor, together with its increased activation, is alone sufficient to result in the androgen-independent growth of LNCaP xenografts. Recent experimental studies that suggest that the posttherapy increase in and continual activation of the androgen receptor are common and crucial features of recurrent prostate cancer provide validation of the model predictions. This approach provides a framework for using mathematical techniques to study novel therapeutic strategies aimed at controlling this disease.

Hierarchical organized tissue structures, with stem cell driven cell differentiation, are critical to the homeostatic maintenance of most tissues, and this underlying cellular architecture is potentially a critical player in the development of a many cancers. Here, we develop a mathematical model of mutation acquisition to investigate how deregulation of the mechanisms preserving stem cell homeostasis contributes to tumor initiation. A novel feature of the model is the inclusion of both extrinsic and intrinsic chemical signaling and interaction with the niche to control stem cell self-renewal. We use the model to simulate the effects of a variety of types and sequences of mutations and then compare and contrast all mutation pathways in order to determine which ones generate cancer cells fastest. The model predicts that the sequence in which mutations occur significantly affects the pace of tumorigenesis. In addition, tumor composition varies for different mutation pathways, so that some sequences generate tumors that are dominated by cancerous cells with all possible mutations, while others are primarily comprised of cells that more closely resemble normal cells with only one or two mutations. We are also able to show that, under certain circumstances, healthy stem cells diminish due to the displacement by mutated cells that have a competitive advantage in the niche. Finally, in the event that all homeostatic regulation is lost, exponential growth of the cancer population occurs in addition to the depletion of normal cells. This model helps to advance our understanding of how mutation acquisition affects mechanisms that influence cell-fate decisions and leads to the initiation of cancers.